• # For Solution

You are given an integer `n`, which indicates that there are `n` courses labeled from `1` to `n`. You are also given a 2D integer array `relations` where `relations[j] = [prevCoursej, nextCoursej]` denotes that course `prevCoursej` has to be completed before course `nextCoursej` (prerequisite relationship). Furthermore, you are given a 0-indexed integer array `time` where `time[i]` denotes how many months it takes to complete the `(i+1)th` course.

## Parallel Courses III solution leetcode

You must find the minimum number of months needed to complete all the courses following these rules:

• You may start taking a course at any time if the prerequisites are met.
• Any number of courses can be taken at the same time.

Return the minimum number of months needed to complete all the courses.

Note: The test cases are generated such that it is possible to complete every course (i.e., the graph is a directed acyclic graph).

Example 1:

## Parallel Courses III solution leetcode

```Input: n = 3, relations = [[1,3],[2,3]], time = [3,2,5]
Output: 8
Explanation: The figure above represents the given graph and the time required to complete each course.
We start course 1 and course 2 simultaneously at month 0.
Course 1 takes 3 months and course 2 takes 2 months to complete respectively.
Thus, the earliest time we can start course 3 is at month 3, and the total time required is 3 + 5 = 8 months.
```

Example 2:

## Parallel Courses III solution leetcode

```Input: n = 5, relations = [[1,5],[2,5],[3,5],[3,4],[4,5]], time = [1,2,3,4,5]
Output: 12
Explanation: The figure above represents the given graph and the time required to complete each course.
You can start courses 1, 2, and 3 at month 0.
You can complete them after 1, 2, and 3 months respectively.
Course 4 can be taken only after course 3 is completed, i.e., after 3 months. It is completed after 3 + 4 = 7 months.
Course 5 can be taken only after courses 1, 2, 3, and 4 have been completed, i.e., after max(1,2,3,7) = 7 months.
Thus, the minimum time needed to complete all the courses is 7 + 5 = 12 months.
```

Constraints:

## Parallel Courses III solution leetcode

• `1 <= n <= 5 * 104`
• `0 <= relations.length <= min(n * (n - 1) / 2, 5 * 104)`
• `relations[j].length == 2`
• `1 <= prevCoursej, nextCoursej <= n`
• `prevCoursej != nextCoursej`
• All the pairs `[prevCoursej, nextCoursej]` are unique.
• `time.length == n`
• `1 <= time[i] <= 104`
• The given graph is a directed acyclic graph.